Letters to the Editor

LETTERS TO THE EDITOR More on Mathematics and the Military Beurling Archive on the Web As a mathematician and a veteran, I read Thomas Gruber’s Arne Beurling1 (1905–1986) was one of the most influen- letter to the editor (January 2018) decrying the influence of tial mathematicians in the area of harmonic and complex the military in mathematics with interest, and I thought it analysis. During WWII Beurling cracked a German cryp- might be useful to add a different perspective to the con- tomachine, the Siemens Geheimschreiber, which was a versation. My own decision to serve was based on a belief long kept secret.2 In 1954 Beurling became Permanent that the dictum Si vis pacem, para bellum1 is correct and Member and Professor at the Institute for Advanced Study. just as relevant now as when first penned. I served with Beurling was very selective in what he published and men and women from all walks of life, of every race, reli- was known for keeping quite a few of his discoveries gion, and background. Their motivation was almost always private. He always required that his work should have selfless: to protect the weak and free the oppressed; to a final elegant form and left a considerable amount of stand watch in the night so that others could sleep with- unpublished notes and manuscripts. Some of them were out fear. I think that our increasingly polarized country included in his Collected Works, but some were not. could learn much from the military; I still remember my Several years ago, Beurling’s grandchildren donated his drill sergeant in basic training saying “The only color in Nachlass to Uppsala University Library. On the suggestion the Army is green,” and it was true. of Lennart Carleson, we selected portions to post at Entering a PhD program after spending 4 years in the https://www.math.uu.se/collaboration/beurling. Army was tough, and brings me to my main point: military They include: service makes returning to graduate school challenging. Many professors are unaware that veterans are the most Beurling’s handwritten notebooks; underrepresented group in academia. Notes of Beurling's seminar talks in Uppsala Before leaving University of Illinois last year, I worked (1938–1951); with the veterans center to set up a mentoring program for Preparatory notes for courses; student veterans. In the course of collecting data, I found Several unpublished manuscripts; that of 1900 tenure stream faculty, only 6 were veterans, Short notes on various topics. and of the 800 faculty in STEM, I was the sole veteran. This Apparently, this material includes results that were is an underrepresentation in STEM faculty by a factor of never published and might be new even now. We believe over fifty times. So I’ll close with a challenge for the AMS that it may be of interest for scholars and students of (and for all of us): How can we bring more of those who complex and harmonic analysis. have served our country into the faculty ranks? —Michael Benedicks —Henry Schenck Royal Institute of Technology, Sweden Iowa State University [email protected] www.math.iastate.edu/hschenck —Mikhail Sodin (Received January 8, 2018) Tel Aviv University [email protected] (Received June 12, 2017) 1 (Benedicks and Sodin) See the obituary by Ahlfors and Carleson, Acta Math. 161 (1988) and the recollections by Ahlfors, Kjellberg, and Wermer in Math. Intelligencer 15:3 (1993). 2 (Benedicks and Sodin) *We invite readers to submit letters to the editor at See Beckman’s Codebreakers: Arne Beurling [email protected]. and the Swedish Crypto Program During World War II, Amer. 1 (Schenck) “If you want peace, prepare for war.” Math. Soc., 2002. MAY 2018 NOTICES OF THE AMS 517 Letters to the Editor Notices Reprint Omits Important Mathematical have been mentioned in connection with Haldane’s work Background of 2016 Nobel Prize in Physics on spin chains: A. Polyakov, Phys. Lett. 59B, 79 (1975). The results in this paper are closely related to the gap We are writing about an article on the 2016 Nobel Prize in the integer spin chain. It is cited in the original article in Physics, which appeared in the Notices of the American of the Academy. Mathematical Society, 64, Number 6, 557–567, (2017). The work for which this prize was awarded has an important II. Papers on Phase Transitions: mathematical component. We thus applaud the decision 4) J. Fröhlich, B. Simon, and T. Spencer, Phase Transi- by the editors of the Notices to publish an article about tions and Continuous Symmetry Breaking, Phys. Rev. Lett. it. However, we are dismayed by some aspects of the 36, 804 (1976). Details appear in: Infrared Bounds, Phase presentation. Transitions and Continuous Symmetry Breaking, Commun. The article reprinted in the Notices was compiled by Math. Phys. 50, 79 (1976). the “Class for Physics of the Royal Swedish Academy of This work contains the first proof of “infrared bounds” Sciences”; (names of authors are not listed). It describes and applies them to prove the existence of phase transi- the groundbreaking work of F. Duncan Haldane, J. Michael tions accompanied by continuous symmetry breaking in Kosterlitz, and David J. Thouless on a “topological phase classical spin systems. transition,” the so-called Kosterlitz-Thouless transition, 5) F. J. Dyson, E. H. Lieb, and B. Simon, Phase transitions and on “topological states of matter.” It was excerpted in quantum spin systems with isotropic and nonisotropic from a longer article authored by the Academy. Unfortu- interactions, J. Stat. Phys. 18, 335 (1978). See also: T. Ken- nately, the Notices chose to selectively include only nine nedy, E. H. Lieb, B. Shastry, Existence of Néel Order in of fifty-five references in the original article. We would like Some Spin-1/2 Heisenberg Antiferromagnets, J. Stat. Phys. to know why the editors of the Notices chose to eliminate 53, 1019 (1988). so many references to important work, and why they did In these papers, symmetry breaking and Néel order not include any references to mathematical results that are established for anti-ferromagnetic quantum magnets. had already strangely been missing in the original article 6) J. Fröhlich and T. Spencer, The Kosterlitz-Thouless released by the Royal Swedish Academy. Transition in the Two-Dimensional Plane Rotator and Cou- We wonder whether the article published in the Notices lomb Gas, Phys. Rev. Lett. 46, 1006 (1981). Details appear was refereed. It appears that knowledgeable mathemati- in: The Kosterlitz-Thouless-Transition in Two-Dimensional cians were not contacted in this matter. Although the Abelian Spin Systems and the Coulomb Gas, Commun. article may be well suited to a physics audience, it neglects Math. Phys. 81, 527 (1981). to mention a significant body of mathematical research This work contains the first rigorous proof of existence closely related to the work of Haldane, Kosterlitz, and of the Kosterlitz-Thouless transition; (it actually settled a Thouless, some of which we have been involved in. We be- controversy on this question). lieve that the editors of the Notices should have consulted A more detailed analysis of this transition appears the mathematical physics community before publication in: P. Falco, Kosterlitz-Thouless Transition Line for the of this article. Addition of a mathematical perspective Two-Dimensional Coulomb Gas, Commun. Math Phys. 312, would have enriched the article and made it more rel- 559–609 (2012). evant for the readership of the Notices. It might also have III. Papers on topological states of matter: inspired further mathematical research. Below, we include 7) D. J. Thouless, Mahito Kohmoto, MP Nightingale, references to some of the important mathematical work and M Den Nijs, Quantized Hall conductance in a two- related to the 2016 Nobel Prize that we feel are useful to dimensional periodic potential, Phys. Rev. Lett. 49, 405 a mathematical readership, (1982). I. Papers on Spin Chains: The relation of this work to homotopy groups of cer- 1) E. H. Lieb, T. D. Schultz, and D. C. Mattis, Two Soluble tain natural vector bundles is pointed out in: J. E. Avron, Models of an Antiferromagnetic Chain, Ann. Phys. 16, 407 R. Seiler, and B. Simon, Homotopy and quantization in (1961). condensed matter physics, Phys. Rev. Lett. 51, 51 (1983). The authors prove that the spin-1/2 chain is gapless. 8) B. Simon, Holonomy, the Quantum Adiabatic Theorem, 2) I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- and Berry’s Phase, Phys. Rev. Lett. 51, 2167 (1983). orous Results on Valence-Bond Ground States in Antifer- This article establishes a connection between Berry’s romagnets, Phys. Rev. Lett. 59, 799 (1987). work and that of Thouless et al. quoted above. This con- This work is mentioned in the Notices, but no reference nection allows the author to use Berry’s ideas to interpret is provided. the integers of Thouless et al. in terms of eigenvalue degeneracies. 3) T. Kennedy and H. Tasaki. Hidden Z2 × Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B, 9) Jean Bellissard, Noncommutative Geometry and 45, 304 (1992). Quantum Hall Effect, in: Proc. of ICM’94, S. D. Chatterji This paper is cited in the original article of the Acad- (ed.), Basel, Boston, Berlin, Birkhäuser Verlag 1995. emy. J. E. Avron, R. Seiler, B. Simon, Charge deficiency, Also, A. Polyakov’s fundamental prediction of the gap charge transport and comparison of dimensions, Com- in the two-dimensional classical Heisenberg model should mun. Math. Phys. 159, 399 (1994). 518 NOTICES OF THE AMS VOLUME 65, NUMBER 5 Letters to the Editor Featured Titles from the Berry’s phase for fermions (which has a quarternionic structure) was studied in: J.

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